# Book:Haïm Brezis/Functional Analysis, Sobolev Spaces and Partial Differential Equations

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## Haïm Brezis:

## Haïm Brezis: *Functional Analysis, Sobolev Spaces and Partial Differential Equations*

Published $2010$, **Springer**

- ISBN 978-0387709130.

### Subject Matter

### Contents

**Preface**

**1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions**

- 1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functionals

- 1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets

- 1.3 The Bidual $E^{**}$. Orthogonality Relations

- 1.4 A Quick Introduction to the Theory of Conjugate Convex Functions

- Comments on Chapter 1

- Exercises for Chapter 1

**2. The Uniform Boundedness Principle and the Closed Graph Theorem**

- 2.1 The Baire Category Theorem

- 2.2 The Uniform Boundedness Principle

- 2.3 The Open Mapping Theorem and the Closed Graph Theorem

- 2.4 Complimentary Subspaces. Right and Left Invertability of Linear Operators

- 2.5 Orthogonality revisited

- 2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint

- 2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators

- Comments on Chapter 2

- Exercises for Chapter 2

**3. Weak Topologies, Reflexive Spaces, Separable Spaces, Uniform Convexity**

- 3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous

- 3.2 Definition and Elementary Properties of the Weak Topology $\map \sigma {E,E^*}$

- 3.3 Weak Topology, Convex Sets and Linear Operators

- 3.4 The Weak* Topology $\map \sigma {E^*,E}$

- 3.5 Reflexive Spaces

- 3.6 Separable Spaces

- 3.7 Uniformly Convex Spaces

- Comments on Chapter 3

- Exercises for Chapter 3

**4. $L^p$ Spaces**

- 4.1 Some Results about Integration That Everyone Must Know

- 4.2 Definition and Elemenary Properties of $L^p$ Spaces

- 4.3 Reflexivity. Separability. Dual of $L^p$

- 4.4 Convolution and regularization

- 4.5 Criterion for Strong Compactness in $L^p$

- Comments on Chapter 4

- Exercises for Chapter 4

**5. Hilbert Spaces**

- 5.1 Definitions and Elementary Properties. Projection onto a Closed Convex Set

- 5.2 The Dual Space of a Hilbert Space

- 5.3 The Theorems of Stampacchia and Lax-Milgram

- 5.4 Hilbert sums. Orthonormal Bases

- Comments on Chapter 5

- Exercises for Chapter 5

**6. Compact Operators, Spectral Decomposition of Self-Adjoint Compact Operators**

- 6.1 Definitions. Elementary Properties. Adjoint

- 6.2 The Riesz-Fredholm Theory

- 6.3 The Spectrum of a Compact Operator

- 6.4 Spectral Decomposition of Self-Adjoint Compact Operators

- Comments on Chapter 6

- Exercises for Chapter 6

**7. The Hille-Yosida Problem**

- 7.1 Definition and Elementary Properties of Maximal Monotone Operators

- 7.2 Solution of the Evolution Problem $\dfrac {\d u} {\d t} + A u = 0$ on $\hointr 0 {+ \infty}$, $\map u 0 = u_0$. Existence and uniqueness

- 7.3 Regularity

- 7.4 The Self-Adjoint Case

- Comments on Chapter 7

**8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension**

- 8.1 Motivation

- 8.2 The Sobolev Space $\map {W^{1,p} } {I}$

- 8.3 The Space $W_0^{1,p}$

- 8.4 Some Examples of Boundary Value Problems

- 8.5 The Maximum Principle

- 8.6 Eigenfunctions and Spectral Decomposition

- Comments on Chapter 8

- Exercises for Chapter 8

**9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in $N$ Dimensions**

- 9.1 Definition and Elementary Properties of the Sobolev Spaces $\map {W^{1,p}} {\Omega}$

- 9.2 Extension Operators

- 9.3 Sobolev Inequalities

- 9.4 The Space $\map {W_0^{1,p} } \Omega$

- 9.5 Variational Formulation of Some Boundary Value Problems

- 9.6 Regularity of Weak Solutions

- 9.7 The Maximum Principle

- 9.8 Eigenfunctions and Spectral Decomposition

- Comments on Chapter 9

**10. Evolution Problems: the Heat Equation and the Wave Equation**

- 10.1 The Heat Equation: Existence, Uniqueness, and Regularity

- 10.2 The Maximum Principle

- 10.3 The Wave Equation

- Comments on Chapter 10

**11. Miscellaneous Complements**

- 11.1 Finite-Dimensional and Finite-Codimensional Spaces

- 11.2 Quotien Spaces

- 11.3 Some Classical Spaces and Sequences

- 11.4 Banach Space over $\C$: What Is Similar and What Is Different

**Solutions of Some Exercises**

**Problems**

**Partial Solutions**

**Notation**

**References**

**Index**